sin θ = 0

How to find the general solution of the equation sin θ = 0?

Prove that the general solution of sin θ = 0 is θ = nπ, n ∈ Z

Solution:

According to the figure, by definition, we have,

Sine function is defined as the ratio of the side opposite divided by the hypotenuse.

Let O be the centre of a unit circle. We know that in unit circle, the length of the circumference is 2π.

If we started from A and moves in anticlockwise direction then at the points A, B, A', B' and A, the arc length travelled are 0, \(\frac{π}{2}\), π, \(\frac{3π}{2}\), and 2π.

Therefore, from the above unit circle it is clear that

sin θ = \(\frac{PM}{OP}\)

Now, sin θ = 0

⇒ \(\frac{PM}{OP}\) = 0

⇒ PM = 0.

So when will the sine be equal to zero?

Clearly, if PM = 0 then the final arm OP of the angle θ coincides with OX or, OX'.

Similarly, the final  arm  OP coincides with OX  or OX'  when θ = 0, π, 2π, 3π, 4π, 5π …………….., -π, , -2π, -3π, -4π, -5π ………., i.e., when  θ = 0  or an integral multiples of π i.e., when θ = nπ where n ∈ Z (i.e., n = 0, ± 1, ± 2, ± 3,…….)

Hence, θ = nπ, n ∈ Z is the general solution of the given equation sin θ = 0


1. Find the general solution of the equation sin 2θ = 0

Solution:

sin 2θ = 0

⇒ 2θ = nπ, where, n = 0, ± 1, ± 2, ± 3,……., [Since, we know that θ = nπ, n ∈ Z is the general solution of the given equation sin θ = 0]

⇒ θ = \(\frac{nπ}{2}\), where, n = 0, ± 1, ± 2, ± 3,…….

Therefore, the general solution of the equation sin 2θ = 0 is θ = \(\frac{nπ}{2}\), where, n = 0, ± 1, ± 2, ± 3,…….


2. Find the general solution of the equation sin \(\frac{3x}{2}\) = 0

Solution:

sin \(\frac{3x}{2}\) = 0

⇒ \(\frac{3x}{2}\) = nπ, where, n = 0, ± 1, ± 2, ± 3,…….[Since, we know that θ = nπ, n ∈ Z is the general solution of the given equation sin θ = 0]

⇒ x = \(\frac{2nπ}{3}\), where, n = 0, ± 1, ± 2, ± 3,…….

Therefore, the general solution of the equation sin \(\frac{3x}{2}\) = 0 is θ = \(\frac{2nπ}{3}\), where, n = 0, ± 1, ± 2, ± 3,…….


3. Find the general solution of the equation tan 3x = tan 2x + tan x

Solution:

tan 3x = tan 2x + tan x

⇒ \(\frac{sin    3x}{cos    3x}\) =  \(\frac{sin  2x}{cos  2x}\) + \(\frac{sin  x}{cos  x}\)

⇒ \(\frac{sin  3x}{cos  3x}\) = \(\frac{sin  2x   cos  x  +  cos  2x   sin  x}{cos  2x    cos  x}\)

cos 3θ sin (2x + x) = sin 3x cos 2x cos x

cos 3x sin 3x = sin 3x cos 2x cosx

cos 3x sin 3x - sin 3x cos 2x cos x = 0

sin 3x [cos (2x + x) - cos 2x cos x] = 0  

sin 3x . sin 2x sin x = 0

Either either, sin 3x = 0 or, sin 2x = 0 or, sin x = 0

3x = nπ or, 2x = nπ or, x = nπ

x = \(\frac{nπ}{3}\)  …..... (1) or, x = \(\frac{nπ}{2}\)  …..... (2) or, x = nπ …..... (3), where n ∈ I

Clearly, the value of x given in (2) are∶ 0, \(\frac{π}{2}\), π, \(\frac{3π}{2}\), 2π, \(\frac{5π}{2}\) ……………., - \(\frac{π}{2}\),- π, - \(\frac{3π}{2}\) , …………

It is readily seen that the solution x = \(\frac{π}{2}\), \(\frac{3π}{2}\), \(\frac{5π}{2}\)………, - \(\frac{π}{2}\), - \(\frac{3π}{2}\),………
Of the above solution do not satisfy the given equation.

Further  to not  that the  rest  solutions of (2) and the  solution of (3) are contained  in the solutions (1).

Therefore, the general solution of the equation tan 3x = tan 2x + tan x is x = \(\frac{3π}{2}\),, where n ∈ I


4. Find the general solution of the equation sin\(^{2}\) 2x = 0

Solution:

sin\(^{2}\) 2x = 0

sin 2x = 0

⇒ 2x = nπ, where, n = 0, ± 1, ± 2, ± 3,……., [Since, we know that θ = nπ, n ∈ Z is the general solution of the given equation sin θ = 0]

⇒ x = \(\frac{nπ}{2}\), where, n = 0, ± 1, ± 2, ± 3,…….

Therefore, the general solution of the equation sin\(^{2}\) 2x = 0 is x = \(\frac{nπ}{2}\), where, n = 0, ± 1, ± 2, ± 3,…….

 Trigonometric Equations






11 and 12 Grade Math

From sin θ = 0 to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.

Share this page: What’s this?

Recent Articles

  1. Intersecting Lines | What Are Intersecting Lines? | Definition

    Jun 14, 24 11:00 AM

    Intersecting Lines
    Two lines that cross each other at a particular point are called intersecting lines. The point where two lines cross is called the point of intersection. In the given figure AB and CD intersect each o…

    Read More

  2. Line-Segment, Ray and Line | Definition of in Line-segment | Symbol

    Jun 14, 24 10:41 AM

    Line-Segment, Ray and Line
    Definition of in Line-segment, ray and line geometry: A line segment is a fixed part of a line. It has two end points. It is named by the end points. In the figure given below end points are A and B…

    Read More

  3. Definition of Points, Lines and Shapes in Geometry | Types & Examples

    Jun 14, 24 09:45 AM

    How Many Points are There?
    Definition of points, lines and shapes in geometry: Point: A point is the fundamental element of geometry. If we put the tip of a pencil on a paper and press it lightly,

    Read More

  4. Subtracting Integers | Subtraction of Integers |Fundamental Operations

    Jun 13, 24 04:32 PM

    Subtraction of Integers
    Subtracting integers is the second operations on integers, among the four fundamental operations on integers. Change the sign of the integer to be subtracted and then add.

    Read More

  5. 6th Grade Worksheet on Whole Numbers |Answer|6th Grade Math Worksheets

    Jun 13, 24 04:17 PM

    6th Grade Worksheet on Whole Numbers
    In 6th Grade Worksheet on Whole Numbers contains various types of questions on whole numbers, successor and predecessor of a number, number line, addition of whole numbers, subtraction of whole number…

    Read More